The study hypothesis is to evaluate the ability of topical treatment with 0.2% brimonidine, a highly selective α2-adrenergic agonist, to impart neuroprotection to the damaged optic nerve in patients with LPG. Comparison is made to 0.5% timolol, a nonselective β-adrenergic antagonist, without reported neuroprotective properties. Patients will be randomly assigned to twice daily double-masked treatment with one of these drugs. Neuroprotection will be assessed by evaluation of automated static visual fields performed at 4 month intervals for 4 years of treatment.